Problem: Is ${930949}$ divisible by $3$ ?
Solution: A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {930949}= &&{9}\cdot100000+ \\&&{3}\cdot10000+ \\&&{0}\cdot1000+ \\&&{9}\cdot100+ \\&&{4}\cdot10+ \\&&{9}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {930949}= &&{9}(99999+1)+ \\&&{3}(9999+1)+ \\&&{0}(999+1)+ \\&&{9}(99+1)+ \\&&{4}(9+1)+ \\&&{9} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {930949}= &&\gray{9\cdot99999}+ \\&&\gray{3\cdot9999}+ \\&&\gray{0\cdot999}+ \\&&\gray{9\cdot99}+ \\&&\gray{4\cdot9}+ \\&& {9}+{3}+{0}+{9}+{4}+{9} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${930949}$ is divisible by $3$ if ${ 9}+{3}+{0}+{9}+{4}+{9}$ is divisible by $3$ Add the digits of ${930949}$ $ {9}+{3}+{0}+{9}+{4}+{9} = {34} $ If ${34}$ is divisible by $3$ , then ${930949}$ must also be divisible by $3$ Add the digits of ${34}$ $ {3}+{4} = \color{#9D38BD}{7} $ If $\color{#9D38BD}{7}$ is divisible by $3$ , then ${34}$ must also be divisible by $3$ $\color{#9D38BD}{7}$ is not divisible by $3$, therefore ${930949}$ must not be divisible by $3$.